Exercises |

E:Prove the following combinatorial principle: IfXandYare finite sets andRis a commutative ring, andj:Y´X -> R, thenå_{fÎYX}Õ_{xÎX}j(f(x),x)=Õ_{xÎX}å_{yÎY}j(y,x).

E:Derive Pólya's theorem directly, using the fact thatfÎYis fixed under^{X}gÎGif and only iffis constant on the cyclic factors ofbar (g).

E:Prove by induction thatå_{pÎSn }q^{l(p)}=[n]!.

E:Derive the formula from exercise by considering a transversal of the left cosets ofS. (Hint: Show that the permutations_{k}ÅS_{n\k}pinSwhich are increasing both on_{n}andkform such a transversal.)n\k

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last changed: August 28, 2001

Exercises |